Metric-Independent Measures for Supersymmetric Extended Object Theories on Curved Backgrounds

For Green-Schwarz superstring sigma-model on curved backgrounds, we introduce a non-metric measure $\Phi \equiv \epsilon^{i j} \epsilon^{I J} (\partial_i \varphi^I) (\partial_j \varphi^J)$ with two scalars $\varphi^I (I = 1, 2)$ used in Two Measure Theory (TMT). As in the flat-background case, the string tension $T= (2 \pi \alpha ' )^{-1}$ emerges as an integration constant for the A_i-field equation. This mechanism is further generalized to supermembrane theory, and to super p-brane theory, both on general curved backgrounds. This shows the universal applications of dynamical measure of TMT to general supersymmetric extended objects on general curved backgrounds.


Introduction
String theory, or more generally, theories of extended objects are believed to be the promising candidates for the unification of all interactions in nature [1]. For such theories, the desiderata is that there should be no fundamental scale involved in their Lagrangians. In other words, the real fundamental theory should involve no fundamental scale in its lagrangian, but instead it should arise at a later stage at the field-equation level, such as after spontaneous symmetry breaking.
One attempt to acquire such a system is found in the so-called 'New Measure' or 'Two-Measure Theory' (TMT). Conventional field theories in curved space time are typically described by actions with the measure d 4 x √ −g which is metric-dependent, but otherwise invariant. However, it is possible to replace such a metric-dependent invariant measure by an alternative metric-independent measure, but is still invariant. Historically, such alternativemeasure theories were first considered by Einstein and Rosen [2].
However, the trouble with (1.2) is that the g ij -field equation yields the unacceptable field equation Φ G mn (∂ i X m )(∂ j X n ) = 0, leaving no dynamical freedom. This problem is solved by an additional term:

the field strength of the abelian gauge field
A i on the 2D world-sheet. The effect of (1.3) is to provide a compensating term proportional to g ij ǫ kl ΦF kl for the g ij -field equation, so the previous term Φ G mn (∂ i X m )(∂ j X n ) does not have to vanish by itself.
The most important conclusion of this bosonic-string formulation [8] is the determination of the string tension T by the field equation of A i as 3) δL where T is an integration constant interpreted as the string tension T = (2πα ′ ) −1 . Despite the presence of the field Φ in (1.2), the original local Weyl symmetry of the action (1.1) is maintained in (1.2), because Φ transforms as a 'scalar-density' like √ −g: where Λ = Λ(σ) is a local parameter. Note that the transformation rule for Φ is also consistent with the solution (1.4). Needless to say, the action I F is also invariant under the Weyl transformation (1.5), because of the special combination (1/ √ −g) ǫ ij Φ.
In ref. [10], this TMT mechanism [6] [7] was further applied to superstring theory [1] in the Green-Schwarz (GS) superstring formulation [11] on the flat background. In our present paper, we consider the GS σ -model on curved 10D superspace background, including unidexterous fermions 4) [12] with fermionic κ -symmetry [13]. Encouraged by the successful application to GS superstring, we further apply similar mechanism to supermembrane theory [14], and further to general super p -brane theories [4] on general curved backgrounds. The application of TMT formulation to bosonic p -brane theories was performed in [10], but not for super p -brane, the simplest case of which is supermembrane with p = 2. In our present paper, we carry out the TMT formulation for these super p -brane with general curved backgrounds.
This paper is organized as follows. In the next section, we present how the dynamical measure for TMT works for GS string σ -model. In section 3, we apply this mechanism to supermembrane theory. Section 4 is for the generalization to super p -branes. Concluding remarks is given in section 5.

GS Superstring σ -Model with Dynamical Measure
Before applying the new measure to GS superstring, we review the fermionic κ -invariance 3) We use the symbol .
= for a field equation or a solution, to be distinguished from algebraic equalities. 4) The meaning of 'unidexterous fermions' will be explained in the second paragraph in the next section.
The field content for the GS superstring σ -model on 10D superspace background [11] [12] is is the 10D curved superspace background coordinates for GS string [11], while ( The action I GS of GS superstring σ -model [11][12] has the string tension T = (2πα ′ ) −1 and the lagrangian , where the local index (i) is used as the subscript [15]. The We are following the superspace notation in [15]. 6) The reason we use the parentheses is to distinguish them from local-coordinate indices. 7) We need the parentheses for (r), (s), ··· to distinguish them from the local curved bosonic index m, n, ···.
The ω i is the 2D Lorentz-connection, which drops out at the lagrangian level. The where the indices (r)(s) = −(s)(r) are for the adjoint representation of SO(32).
The action I GS is invariant under the fermionic κ -symmetry transformation [13][12] [1]: We give here the explicit total divergence form for δ κ L GS that will be useful later: leading to the invariance δ κ I (0) As for the concept of 'general backgrounds', we add the following clarification. 'General backgrounds' imply that at least 10D space-time is curved by gravity with the non-trivial 10D metric g mn . However, once gravity is introduced, for the consistency of the system with supersymmetry, all other supersymmetric partner superfields should be also introduced in a way consistent with N = 1 local supersymmetry in 10D. In other words, all 10D background superfields should be introduced consistently. They are not just limited to the NS-NS fields g mn , B mn and ϕ. To be more specific, the θ = 0 components corresponding to 10D component fields [16] are listed as (e a m , ψ a α , B ab , χ α , ϕ, A a (r)(s) , λ α (r)(s) ).
Once we have established (2.4) for the conventional Green-Schwarz σ -model [11][12], it is straightforward to confirm the κ -invariance of our new action with the new measure consisting of scalar fields ϕ I in place of the conventional measure from the metric.
To this end, we enlarge the field content to ( Here the new scalar field ϕ I has the index I = 1, 2, and A i is an Abelian vector field whose field strength As is already known in the bosonic string case [8], a term linear in F ij is needed to cancel the unwanted term in the V (i) j -field equations. Moreover, this term is also needed from the viewpoint of κ -invariance of the total action, as will be seen next.
We propose our total action I GS ≡ d 2 σ L GS to be Our action I GS is invariant under the fermionic κ -transformation rule The invariance δ κ I GS = 0 is confirmed as follows. First, δ κ Φ = 0 and δ κ V = 0 lead to δ κ χ = 0, drastically simplifying the whole computation. This is because the variation δ κ L is only from δ κ L GS and Φ δ κ F ++,−− . In particular, we already know the former as in (2.4). After a partial integration, the former yields a derivative on χ, which is cancelled by the variation δ κ F ij again after a partial integration. Note that the invariance δ κ I GS = 0 is not approximated one, such as only up to certain degrees in terms of ψ + (r) . In other words, our action I GS is confirmed to be κ -invariant to all orders. Thus we conclude that there is no problem for the κ -invariance of our action: δ κ I GS = 0.
We next study all the field equations of A i , ψ + (r) , V ++ i , V −− i and ϕ I in turn: This is the simplest one derived as This implies that the combination V Φ is a constant T , i.e., where the constant T is interpreted as the string tension T = (2πα ′ ) −1 .
(ii) ψ + (r) -Field Equation: The direct computation gives To get the last expression, have used (2.8).
(iii) V ++ i -Field Equation: The direct variation yields This equation yields, when multiplied by respectively V −− i and V ++ i , The former is nothing but the conventional Virasoro condition, while the latter fixes the value of the new field strength F ++,−− . 8) This situation is parallel to the bosonic case [8].
The direct variation yields When multiplied by V −− i and V ++ i , eq. (2.13) yields respectively The former is nothing but the usual Virasoro condition with the unidexterous fermions, while the latter is consistent with (2.12b), as desired.
(v) ϕ I -Field Equation: The direct computation gives This further yields
due to the last term in (2.15) vanishing upon the ψ -field equation (2.10), while M is a real integration constant. In our present TMT applied to GS superstring, or TMT applied to bosonic string [8], this constant M is fixed to be zero, because of V ++ i and V −− i -field equations (2.12a) and (2.14a). This situation is different from more general TMT formulations [6][7], in which the constant M remains to be non-zero in general.
To summarize, our system has the same field equations as the conventional GS superstring  [12]. This situation is parallel to the aforementioned bosonic string [8] in the Polyakov-type formulation [9], and the GS superstring flat-background case [10].
We mention the fact that the equivalence between I (0) GS for conventional GS [11][12] and our TMT generalization I GS is valid only at the classical level. Even for the conventional GS formulation [11] [12], quantum computations are limited for general curved backgrounds, such as sigma-model β -function computations [17]. Since the quantum-level computations are highly non-trivial and need more arrangements for computations, it is beyond the scope of our present paper.
Even though TMT formulations for superstring were presented for flat background in [10], the importance here is that we have confirmed it also for GS superstring with general curved 10D superspace backgrounds [12].
The action I where he T is the membrane tension, while the Π's represents the superspace pull-back [18]. 9) 9) We use the notation in [15] in superspace.

The action I
(0) SM is invariant under the fermionic κ -symmetry transformation rule [13] δ The explicit form of the variation δ κ L (0) SM with surface term included will be useful for later purpose: After using the relationships also known as the 'embedding condition'. We also use the 11D superspace constraints [18] T αβ c = +i(γ c ) αβ , G αβcd = + 1 2 (γ cd ) αβ . (3.7) Our field content of TMT [6][7] applied to supermembrane [14] is (Z M , g ij , ϕ I , A i IJ , C ij ).
Here the scalar ϕ I (I = 1 Our action I SM ≡ d 3 σ L SM has the lagrangian Other than the presence of SO(3) -minimal couplings, this form is parallel to the scalardensity function used in TMT [6] [7].
We next confirm the consistency of the field equations of our fields: (C ij , A i IJ , ϕ I , g ij , Z M ): The consequence of this simplest field equation is important: This means that the membrane tension T emerges as the integration constant for the C ij -field equation, as one of our desired objectives.
(ii) The A i IJ -Field Equation: δL δL In the usual TMT formulation [6][7], the covariant derivative D i is the ordinary derivative ∂ i , so that it commutes with P j I P k K . Eventually, the square bracket of (3.12) should be an arbitrary real constant M [6] [7]. However, the crucial difference here is that D i does not commute with the factor P j I P k K , so that the square bracket in (3.12) is not necessarily an arbitrary constant. Fortunately, the A i IJ -field equation (3.11) provides a stronger condition, such that the content of the square bracket in (3.12) vanishes. This is the advantage of the minimal coupling of the SO(3) -gauge field A i I in our system.
(iv) The g ij -Field Equation: This equation is the most crucial test, because we need the embedding condition g ij .
= Π i a Π ja [14]. In fact, we get which is further simplified under (3.11) as (3.14) When the trace of this equation (Π i a ) 2 .
This field equation is eventually the same as in the supermembrane theory [14]: For reaching this final form, we have used the lemma (3.5), and the basic relationship with the 11D Lorentz connection superfield φ DB A for an arbitrary variation δE A ≡ (δZ M )E M A . These field equations coincide with those in conventional supermembrane theory [14], and provide the supporting evidence of the consistency of our total system. Note that our lagrangian (3.8) is reduced to the conventional supermembrane lagrangian The explicit form of our fermionic κ -transformation rule is while we do not specify δ κ g ij in our 1.5-order formalism, for the same reason already mentioned. Keeping this point in mind, and also using the result (3.4), we get the κ -invariance of our action where (P −1 ) J i is the inverse matrix of P i I , satisfying (P −1 ) J i P i I = +δ J I , and the first equality . = in (3.19) symbolizes the usage of g ij .
= Π i a Π ja and a surface integration.
We have thus confirmed the invariance of our action δ κ I SM = 0 under the fermionic κ -transformation (3.18) with general curved 11D backgrounds.

Generalization to Super p -Branes
Once we have understood the case of supermembrane, the generalization to super p -branes (p ≥ 3) [4] is rather straightforward. For such a general ∀ p -brane formulation, the previous supermembrane for p = 2 becomes just the special case with the superspace [18] for 11D target space-time.
Our total action is I pB ≡ d p+1 σ L pB , where with d ≡ p + 1 and The fermionic κ -transformation rule is As is easily seen, the previous supermembrane case is the special case of p = 2.
Even though TMT formulation was presented in [10] for super p -branes [4] for flat backgrounds, our present result is valid for general curved backgrounds in the target space-time.

Concluding Remarks
In this paper, we have applied TMT [6][7] to GS superstring [1] on general curved backgrounds, carrying out the objective to generate the superstring tension T only as an integration constant, while it is absent from the fundamental lagrangian. This mechanism is further applied to supermembrane [14], and super p -brane theories [4], both on general curved backgrounds. The lagrangian of GS superstring is (2.6) with κ -invariance (2.7), that of supermembrane is (3.8) with κ -invariance (3.18), and that of super p -brane is (4.1) with κ -invariance (4.3).
The new feature of our result compared with [10] is that κ -invariances with TMT dynamical measures have been confirmed for supersymmetric extended objects, such as supermembrane, and more general super p -branes on general curved backgrounds. Even for GS superstring, we have added unidexterous fermions which were not treated in [10]. Even though the extra factor χ is multiplied by the conventional super p -brane lagrangian [14], all new contributions are cancelled by δ κ C ij and δ κ A i IJ .
In principle, we can apply TMT formulations [6][7] to Dp -brane theory [19] in a similar fashion. In such a case, we need world-sheet Born-Infeld vectors. In practice, however, the required computation will be more involved beyond the scope of this Letter. We leave such formulations for future projects.
Our present results show that the dynamical measure in TMT [6][7] has general universal features applicable to supersymmetric extended objects, such as GS superstring [1], supermembrane [14], and super p -branes [4], on general curved backgrounds. Even though the generalizations to supersymmetric extended objects on general curved backgrounds seem straightforward, we have to confirm this conjecture by explicit computations. Based on our encouraging results, it is natural to expect that the basic properties of TMT dynamical measure [6] [7] are universally applicable to even other (supersymmetric) extended objects.